I've done some research into partitions and am yet to find any resources to understand the following:
Given a number $n$ and the restrictions that:
- Using only the numbers $1, 2, ..., m$;
- A maximum of $p$ times per number.
How many ways are there to add together these numbers to be less than or equal to $n$.
As an example using $n = 5, m = 3,$ and $p = 2$.
You can find:
- $1$
- $1 + 1, 2$
- $2 + 1, 3$ (not $1 + 1 + 1$)
- $2 + 1 + 1, 2 + 2, 3 + 1$
- $3 + 1 + 1, 2 + 2 + 1, 3 + 2$
so the answer here would be $1 + 2 + 2 + 3 + 3 = 11$
If someone could point me in the right direction, that would be very helpful.
Thanks.